Mesh and Time Step Sensitivity Studies

Assignment :

Go to the ANGEL web site and complete Reading
Assignment 1.
Come prepared to discuss what you've read.
Start HW5

Most rigorous studies to quantify error associated with the selected
mesh or
time step sizes are based on Richardson Extrapolation. This
started as a
means of improving the accuracy of numerical solutions to differential
equations, but also can be used as a basis for estimating errors
associated
with selection of the mesh and time step size. Without
understanding
these errors, speculation on the quality of various physical models
associated
with a reactor safety code is on shaky ground. If your mesh is
reasonably
fine, and you know the order of accuracy of your
method,
you can use Richardson Extrapolation and results from two different
grids (or
two different time steps) to say something about the error. If
you don't
know the methods accuracy, or don't have confidence that your mesh is
fine
enough, you need a study with at least three different spatial
divisions (or
time step sizes).

Mesh
and time step sensitivity studies lead to an estimate of error
associated with
discretization, and are also important in procedures used to detect
software
errors. Roache and Oberkampf
have good discussions of this error analysis. It basically boils
down to
fitting a curve to a sequence of results and extrapolating beyond those
results
to estimate the limiting answer with zero mesh length or time
step.
Consider a sequence of three mesh lengths or time step sizes (from
smallest to
largest) h1, h2,
and h3. Normally the sequence is generated with a
constant
refinement ratio:

r=
h3/ h2= h2/h1.

Let
f1, f2, and
f3 be the computed results
at the same
point in space and time for the three corresponding values of h.
Taking a
clue from truncation error analysis, we look for an expression for f as
a
function of h in the form:

f(h) = fexact + a hp

hence,

su

btracting
the equations in pairs gives

hence

Note
that if the scaling ratio r is constant not constant (h3/h2
is not equal to h2/h1), we can solve for p, but
it is much more
difficult. Also notice that if values of f are not
monotonic, the
formula won’t work. Although it is possible to have non-monotonic
convergence, you will need results on more than three grids (or time
steps) to
convince me of any error estimate in such situations.

Given
a value of p, equations for the two finest meshes can be solved for the
remaining unknowns.

As
a result the error on the finest mesh can be estimated as:

Note
that if you have faith in the value of p obtained from a Taylor
series truncation error analysis, you can use this expression with
results from
just two meshes to give an error estimate. However, this is a
dangerous
approach. With just two meshes (or time step sizes) you can’t
always be certain that your spacing is small enough that higher order
terms in
the Taylor
expansion are insignificant.

These
formulas for error and order of accuracy are relatively easy to
implement for
time step sensitivity and finite difference mesh sensitivity studies
where the
refined grids contain the points evaluated on the coarser grids.
However,
for finite volume, if I double the number of volumes, the volume
centers
don’t match between two levels of refinement. Since f1 and f2
must
be compared at the same points in space and time interpolation is
required on
one of the grids. Be careful that your interpolation is
sufficiently
accurate that the calculated value of p tells you about the order of
accuracy
of your finite volume approximation rather than the order of accuracy
of your
interpolator.

Roache
notes that the above equation is not always a reliable bound on
error. He
recommends multiplying any such error estimate by a “Factor of
Safety” (Fs). Values of this factor would range from a
high of 3 for a two mesh study to a low of 1.25 for a three mesh study
confirming convergence of the mesh. Use of 3 corresponds to
replacement
of an error estimate for a second order method with one for a first
order
method. Roache also recommends reporting of error in terms of a
Grid
Convergence Index (GCI):